Propagation phenomena for a bistable Lotka–Volterra competition system with advection in a periodic habitat

Abstract

This paper is concerned with a Lotka–Volterra competition system with advection in a periodic habitat

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_1}{\partial t}=d_1(x)\frac{\partial ^2 u_1}{\partial x^2}-a_1(x)\frac{\partial u_1}{\partial x} +u_1\left( b_1(x)-a_{11}(x)u_1-a_{12}(x)u_2\right) ,\\ \frac{\partial u_2}{\partial t}=d_2(x)\frac{\partial ^2 u_2}{\partial x^2}-a_2(x)\frac{\partial u_2}{\partial x} +u_2\left( b_2(x)-a_{21}(x)u_1-a_{22}(x)u_2\right) , \end{array}\right. } t>0,~x\in {\mathbb {R}}, \end{aligned}$$

where $d_i(\cdot )$, $a_i(\cdot )$, $b_i(\cdot )$, $a_{ij}(\cdot )$ $(i,j=1,2)$ are L-periodic functions in $C^\nu ({\mathbb {R}})$ for some $L>0$. Under certain assumptions, the system is bistable between two (linearly) stable periodic steady state solutions $(0,u_2^*(x))$ and $(u_1^*(x),0)$. We establish the existence of pulsating traveling front $(U_1(x,x+ct),U_2(x,x+ct))$ connecting $(0,u_2^*(x))$ to $(u_1^*(x),0)$. Furthermore, we confirm that the pulsating traveling front is globally asymptotically stable for wave-like initial values. We finally show that such pulsating traveling front is unique (up to translation). The methods involve the sub- and supersolutions, spreading speeds of monostable systems, and the monotone semiflows approach. From the biological point of view, this kind of pulsating traveling front provides a spreading way for two strongly competing species interacting in a heterogeneous habitat.